If $\tan 25^{\circ} = x,$ then, $then \frac{\tan 155^{\circ} - \tan 115^{\circ}}{1 + \tan 155^{\circ} \tan 115^{\circ}}$ is equal is

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$\frac{1 + x^{2}}{2 x}$

$\sqrt[]{\frac{1 + x^{2}}{1 - x^{2}}}$

$\frac{1 - x^{2}}{2 x}$

$\frac{1 - x^{2}}{1 + x^{2}}$

answer is A.

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Detailed Solution

$\frac{\tan 155^{\circ} - \tan 115^{\circ}}{1 + \tan 155^{\circ} \tan 115^{\circ}}$

$= \frac{\tan (180^{\circ} - 25^{\circ}) - \tan (90^{\circ} + 25^{\circ})}{1 + \tan (180^{\circ} - 25^{\circ}) \tan (90^{\circ} + 25^{\circ})}$

$= \frac{- \tan 25^{\circ} + \cot 25^{\circ}}{1 + \tan 25^{\circ} \cot 25^{\circ}} = \frac{1}{2} (- x + \frac{1}{x}) = \frac{1 - x^{2}}{2 x}$

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